Some examples of random walks on free products of discrete groups

Summary We consider the random walk (X_n) associated with a probability p on a free product of discrete groups. Knowledge of the resolvent (or Green's function) of p yields theorems about the asymptotic behaviour of the n-step transition probabilities p^*n(x)=P(X_n= x¦ X₀=e) as n. Woess [15], Cartwright and Soardi [3] and others have shown that under quite general conditions there is behaviour of the type p^*n(x)C_x^– n n^– 3/2. Here we show on the other hand that if G is a free product of m copies ofZ ^r and if (X_n) is the « average » of the classical nearest neighbour random walk on each of the factorsZ ^r, then while it satisfies an « n^–3/2 — law » for r small relative to m, it switches to an n^– r/2 -law for large r. Using the same techniques, we give examples of irreducible probabilities (of infinite support) on the free groupZ ^*m which satisfyn ^– for .     

Characterization of a Nudix hydrolase from <Emphasis Type="Italic">Deinococcus radiodurans</Emphasis> with a marked specificity for (deoxy)ribonucleoside 5'-diphosphates

Background  

Nudix hydrolases form a protein family whose function is to hydrolyse intracellular nucleotides and so regulate their levels and eliminate potentially toxic derivatives. The genome of the radioresistant bacterium Deinococcus radiodurans encodes 25 nudix hydrolases, an unexpectedly large number. These may contribute to radioresistance by removing mutagenic oxidised and otherwise damaged nucleotides. Characterisation of these hydrolases is necessary to understand the reason for their presence. Here, we report the cloning and characterisation of the DR0975 gene product, a nudix hydrolase that appears to be unique to this organism.     

Calculations of the electronic structures of cage molecules using free-electron orbitals as a basis

A non-empirical molecular orbital method, particularly suitable for calculations on cage-like molecules, is described. The method uses as basis functions the set of free-electron functions which are the solutions of Schrödinger's equation for an electron confined between two concentric, spherical potential energy barriers. Application of the theory to the SCF calculation of the energies of the delocalized electrons in benzene and tetrasulphur tetranitride shows that the model is capable of interpreting the properties of such systems. However, it does highlight a difficulty in the calculation of excited state energies with one-centre models which appears to be largely unrecognized.Extension of the method to a consideration of all the valence electrons, using P₄ as an example, reveals problems the origin of which is an inadequate treatment of the core electrons. It is suggested that these problems may best be dealt with by use of a suitable pseudo potential.     

Organophotoredox/palladium dual catalytic decarboxylative Csp3–Csp3 coupling of carboxylic acids and π-electrophiles

A dual catalytic decarboxylative allylation and benzylation method for the construction of new C(sp³)–C(sp³) bonds between readily available carboxylic acids and functionally diverse carbonate electrophiles has been developed. The new process is mild, operationally simple, and has greatly improved upon the efficiency and generality of previous methodology. In addition, new insights into the reaction mechanism have been realized and provide further understanding of the harnessed reactivity.

A dual catalytic decarboxylative allylation and benzylation method for the construction of new C(sp³)–C(sp³) bonds between readily available carboxylic acids and functionally diverse carbonate electrophiles has been developed.     

Imidazopyridine and pyrimidinopyridine systems from perfluorinated pyridine derivatives

Annelation of pentafluoropyridine via an intramolecular nucleophilic aromatic substitution process with benzamidine gave an imidazopyridine system in high yield in a two step process whilst alkyl amidines gave 4-aminotetrafluoropyridine by a competing elimination reaction. 4-Phenylsulfonyl tetrafluoropyridine reacts with amidines to give the corresponding imidazo[4,5-b]pyridine systems. In contrast, 4-cyanotetrafluoropyridine gave a [6,6]-fused pyrimidinopyridine system arising from initial nucleophilic substitution at the C-3 position of the pyridine ring followed by intramolecular cyclization onto the pendant cyano group. The systems prepared by this annelation methodology further demonstrate the utility of perfluorinated heterocyclic substrates for the synthesis of heterocyclic scaffolds that possess multiple functionality and have potential applications in the drug discovery arena.     

The heat of formation of sulphuryl fluoride and the stability of fluorosulphates

A standard enthalphy of -767.2 kJ mol^-1 was determined for sulphuryl fluoride from the heat of alkaline hydrolysis, and a standard free energy of -720.8 kJ mol^-1 derived. The thermodynamically preferred product of ionic fluorosulphate decompositions is always sulphuryl fluoride. Kinetic control must predominate when sulphur trioxide is the main product.     

Photoredox/Cobalt Dual-Catalyzed Decarboxylative Elimination of Carboxylic Acids: Development and Mechanistic Insight

Recently, dual-catalytic strategies towards the decarboxylative elimination of carboxylic acids have gained attention. Our lab previously reported a photoredox/cobaloxime dual catalytic method that allows the synthesis of enamides and enecarbamates directly from N-acyl amino acids and avoids the use of any stoichiometric reagents. Further development, detailed herein, has improved upon this transformation's utility and further experimentation has provided new insights into the reaction mechanism. These new developments and insights are anticipated to aid in the expansion of photoredox/cobalt dual-catalytic systems.     

Calcium Carbonate Polyamorphism and Its Role in Biomineralization: How Many Amorphous Calcium Carbonates Are There?

Although the polymorphism of calcium carbonate is well known, and its polymorphs—calcite, aragonite, and vaterite—have been highly studied in the context of biomineralization, polyamorphism is a much more recently discovered phenomenon, and the existence of more than one amorphous phase of calcium carbonate in biominerals has only very recently been understood. Here we summarize what is known about polyamorphism in calcium carbonate as well as what is understood about the role of amorphous calcium carbonate in biominerals. We show that consideration of the amorphous forms of calcium carbonate within the physical notion of polyamorphism leads to new insights when it comes to the mechanisms by which polymorphic structures can evolve in the first place. This not only has implications for our understanding of biomineralization, but also of the means by which crystallization may be controlled in medical, pharmaceutical, and industrial contexts.     

Noise- and inertia-induced inhomogeneity in the distribution of small particles in fluid flows

The dynamics of small spherical neutrally buoyant particulate impurities immersed in a two-dimensional fluid flow are known to lead to particle accumulation in the regions of the flow in which vorticity dominates over strain, provided that the Stokes number of the particles is sufficiently small. If the flow is viewed as a Hamiltonian dynamical system, it can be seen that the accumulations occur in the nonchaotic parts of the phase space: the Kolmogorov-Arnold-Moser tori. This has suggested a generalization of these dynamics to Hamiltonian maps, dubbed a bailout embedding. In this paper we use a bailout embedding of the standard map to mimic the dynamics of neutrally buoyant impurities subject not only to drag but also to fluctuating forces modeled as white noise. We find that the generation of inhomogeneities associated with the separation of particle from fluid trajectories is enhanced by the presence of noise, so that they appear in much broader ranges of the Stokes number than those allowing spontaneous separation. (c) 2002 American Institute of Physics.